Atomic Routing Games on Maximum Congestion

Atomic Routing Games on Maximum Congestion Costas Busch and Malik Magdon-Ismail Rensselaer Polytechnic Institute, Dept. of Computer Science, Troy, NY 12180, USA. {buschc,magdon}@cs.rpi.edu Abstract. We study atomic routing games on networks in which players choose a path with the objective of minimizing the maximum congestion along the edges of their path. The social cost is the global maximum congestion over all edges in the network. We show that the price of stability is 1. The price of anarchy, P oa, is determined by topological properties of the network. In particular, P oa = O(l + log n), where l is the length of the longest path in the player strategy sets, and n is the size of the network. Further, κ 1 P oa c(κ 2 + log 2 n), where κ is the length of the longest cycle in the network, and c is a constant. 1 Introduction A fundamental issue in the management of large scale communication networks is to route the packet traffic so as to optimize the network performance. Our measure of network performance is the worst bottleneck (most used link) in the system. We model network traffic as finite, unsplittable packets (atomic flow) [22, 26], where each packet s path is controlled independently by a selfish player. The Nash equilibrium (NE) is a natural outcome for a game with selfish players a stable state in which no player can unilaterally improve her situation. In the recent literature, the price of anarchy (P oa) [15, 24] and the price of stability (P os) [1, 2] have become prevalent measures of the quality of the equilibria of uncoordinated selfish behavior relative to coordinated optimal behavior. The former quantifies the worst possible outcome with selfish agents, and the latter measures the minimum penalty in performance required to ensure a stable equilibrium outcome. We study routing games with N players corresponding to N sourcedestination pairs of nodes on a network G. The strategy set available to each player is a set of edge-simple paths from the player s source to the destination (typically the strategy set consists of all edge-simple paths in G). A pure strategy profile is a selection of a single path (strategy) by each player from her respective strategy set. We study pure Nash equilibria. In our context, a pure strategy profile corresponds to a routing p, a collection of paths, one for each player. We refer to Nash equilibra in this context as Nash-routings. A routing p causes congestion in the network: the congestion C e on an edge e is the number of paths in p that use this edge; the congestion C pi of a path p i p is the maximum congestion over all edges on the path; the congestion C of the network is

the maximum congestion over all edges in the network. The dilation D is the maximum path length in p. Since a packet is to be delivered along each player s path, a natural choice for social cost is the maximum delay incurred by a packet. The packets can be scheduled along the paths in p with maximum delay O(C + D) [6, 17, 18, 23, 25]. In heavily congested networks, C D, and the maximum delay of a packet is governed by the congestion C. Thus, the network congestion is an appropriate social cost this choice for the social cost is often referred to as the maximum social cost [4, 5, 15, 27]. Consider player i with path p i p. It is shown in [3] that player i s packet can be delivered in time Õ(C p i + p i ), where p i is the path length (this holds for all players simultaneously). In congested networks, C pi p i, and so it is appropriate to use C pi as the player cost, along her chosen path. This choice of player cost is typically referred to as the maximum player cost. The maximum player cost is is appropriate since this is what governs the delay experienced by that player in a highly congested network [3]. In the literature it is common to use the sum player cost (instead of the maximum) [7, 9, 14, 15, 21, 27]. However, the sum of congestions does not govern the packet delays, since when a packet waits for a particular congested edge to clear of other packets, the other congested edges in its path can be cleared simultaneously. It is the maximum player and social costs that are appropriate metrics for atomic routing games. 1.1 Contributions We give the first comprehensive analysis of routing games with maximum player and social cost. We study the quality of pure Nash-routings with respect to the price of stability and anarchy. In our first result, we establish that there exist optimal Nash-routings where the social cost (congestion) is equal to the optimal coordinated cost; in other words, P os = 1 (the price of stability expresses the ratio of the optimal social cost in the Nash-routing with the optimal coordinated cost). We also show that any best response dynamic, a sequence of best response moves of players, converges to a Nash-routing in a finite amount of time. Thus, we can easily obtain Nash-routings, starting from arbitrary initial routings. Theorem 1. For every routing game: (i) There is a pure Nash-routing which is optimal (P os = 1). (ii) Every best response dynamic converges to a Nash-routing in finite time. We continue by examining the quality of the worst case Nash-routings. The price of anarchy, P oa, expresses the ratio of the social cost in the worst-case Nash-routing to the optimal coordinated cost. We bound the price of anarchy in terms of topological properties of the network. The next result bounds the price of anarchy for arbitrary instances of routing games in terms of the maximum path-lengths in the strategy sets: 2

Theorem 2. For any routing game where the strategy sets of the payers have paths with length at most l, P oa < 2(l + log n). Theorem 2 gives good bounds for the price of anarchy for networks where it is natural to use paths with short length. For example in the Hypercube and Butterfly [16], if we choose bit-fixing paths, then l = O(log n), which implies that P oa c log n, for some constant c. Our next result characterizes the worst case Nash-routing in terms of the longest cycle of the network. For a graph G, the edge-cycle number κ e (G) is the length of the longest edge simple cycle in G; we will drop the dependence on G when the context is clear. Theorem 3. For any undirected graph G with edge-cycle number κ e, (i) there exists a routing game for which P oa κ e 1; (ii) for any routing game, P oa c(κ e 2 + log 2 n), for some constant c. Let m denote the number of edges in the network. Since κ e m, we have that P oa c m 2. In graphs with Euler cycles, κ e = m. Therefore, Theorem 3 implies that m 1 P oa c m 2 (we use c to represent a generic constant). The lower bound of Theorem 3 (part i) is obtained by constructing a game instance where the players have their sources and destination on the largest cycle. To prove the upper bound of Theorem 3 (part ii), we use Theorem 2. For 2-connected graphs, every pair of nodes has two edge-disjoint paths connecting them (Menger s theorem [32]), from which we establish that l c κ e 2. The cycle upper bound follows immediately by using Theorem 2. If the graph G is not 2-connected, then the relation l c κ e 2 may not hold. To obtain the result for a general graph G, we decompose G into a tree of 2-connected components. We show that if in G the Nash-routing has network congestion C, then there is some 2-connected component G which has congestion C C. At the same time the players in G are in a partial Nash-routing, where many of them are locally optimal. A generalization of Theorem 2 to partial Nash-routings, helps to establish the upper-bound of Theorem 3. 1.2 Related Work General congestion games were introduced and studied in [22, 26]. The application of game theory in computer science, specifically the introduction of the price of anarchy was introduced in [15]. Since then, many models have been studied, categorized by: the topology of the network; the nature of the player and social costs; the nature of the traffic (atomic or splittable); the nature of the strategy sets; the nature of the equilibria studied (pure or mixed). A brief taxonomy of some relevant existing results, according to the kind of flow (atomic or splittable) and equilibria (mixed or pure), and according to the social cost SC and player cost pc (sum or maximum), are shown in the following two tables. Atomic Flow Splittable Flow Pure [4, 19, 26], [31], Our Work [27 30] Mixed [7 9, 11 15, 20, 21, 24] [5], [10] 3

Max SC Sum SC Other SC ** Max pc Our Work [19] Sum pc [4, 5, 27] [7 11, 14, 15, 21, 24] [4, 28 30], [13, 31] [12, 20] [19, 26] ( : A Specific network model is used, eg. parallel links, or specific player strategy sets, eg. singleton sets. : Results on existence or convergence to equilibrium, as opposed to quality of equilibria). Typically, the research in the literature has focused on computing upper and lower bounds on the price of anarchy. The vast majority of the work on maximum social cost has been for parallel link networks, with only a few recent results on general topologies [4, 5, 27]. Essentially, all of the work has focused on the sum player cost, which corresponds to the sum of the edge congestions on a path (as opposed to the maximum edge congestion on the path, which we consider here). The only result which has a brief discussion of the maximum player cost is [19] where the authors focus on parallel link networks, but also give some results for general topologies. In [19], the main content is to establish the existence of pure Nash-routings. We present a systematic study of pure Nash-routings in atomic routing games. Pure equilibria with atomic players and maximum player cost introduces essentially combinatoric conditions for the equilibria, in contrast to infinitelly splittable flow, or mixed equilibria, which can be characterised by Wardrop-type equilibrium conditions. Outline of Paper. In Section 2 we give some basic definitions. We prove Theorem 1 in Section 3. We continue with the proof of Theorem 2 in Section 4. The lower bound of Theorem 3 is proven in Section 5. In the same section we prove the upper bound of Theorem 3 for 2-connected graphs. We give the general version of the upper bound in Section 6. We conclude in Section 7. Some of the technical proofs have been omitted for space considerations, and will be presented in a full version of this paper. 2 Definitions An instance R of a routing (congestion) game is a tuple (N, G, {P i } i N ), where N = {1, 2,..., N} are the players, G = (V, E) is an undirected connected graph with V = n, and P i is a collection of edge-simple paths. Each path in P i is a path in G that has the same source s i V and destination t i V ; each path in P i is a pure strategy available to player i. A pure strategy profile p = [p 1, p 2,, p N ] is a collection of pure strategies (paths), one for each player, where p i P i. We refer to a pure strategy profile as a routing. On a finite network, a routing game is necessarily a finite game. For any routing p and any edge e E, the edge-congestion C e (p) is the number of paths in p that use edge e. For any path p, the path-congestion C p (p) is the maximum edge congestion over all edges in p, C p (p) = max e p C e (p). The network congestion is the maximum edge-congestion over all edges in E, C(p) = max e E C e (p). The social or global cost SC(p) is the network congestion, SC(p) = C(p). The player or local cost pc i (p) for player i is her path- 4

congestion, pc i (p) = C pi (p). When the context is clear, we will drop the dependence on p and use C e, C p, C, SC, pc i. We use the standard notation p i to refer to the collection of paths {p 1,, p i 1, p i+1,, p N }, and (p i ; p i ) as an alternative notation for p which emphasizes the dependence on p i. Player i is locally optimal in routing p if pc i (p) pc i (p i ; p i) for all paths p i P i. A routing p is in a Nash Equilibrium (p is a Nash-routing) if every player is locally optimal. Nash-routings quantify the notion of a stable selfish outcome. A routing p is an optimal pure strategy profile if it has minimum attainable social cost: for any other pure strategy profile p, SC(p ) SC(p). We quantify the quality and diversity of the Nash-routings by the price of stability (P os) and the price of anarchy (P oa) (sometimes referred to as the coordination ratio). Let P denote the set of distinct Nash-routings, and let SC denote the social cost of an optimal routing p. Then, P os = inf p P SC(p) SC(p), P oa = sup SC p P SC. 3 Existence of Optimal Nash-routings The goal in this section is to establish Main Theorem 1. For routing p, the congestion vector C(p) = [m 0 (p), m 1 (p), m 2 (p),...], where each component m k (p) is the number of edges with congestion k. Note that k m k(p) = m, where m is the number of edges in the network. The social cost (network congestion) SC(p) is the maximum k for which m k > 0. We define a lexicographic total order on routings as follows. Let p and p be two routings, with C(p) = [m 0, m 1, m 2,...], and C(p ) = [m 0, m 1, m 2...]. Two routings are equal, written p= c p, if and only if m k = m k for all k 0; p< c p if and only if there is some k such that m k < m k and k > k, m k m k. Let (N, G, {P i } i N ) be an instance of a routing game. Since there are only finitely many routings (as a player s path may use any edge at most once), there exists at least one minimum routing w.r.t. the total order < c. There may be many distinct routings all of which are minimum (and equal to each other). Let p be a minimum routing (which exists); then, for all routings p, p c p. Every minimum routing is optimal; indeed, if SC(p) < SC(p ) for some other routing p, then the maximum k for which m k (p) > 0 is smaller than the corresponding k for p, contradicting the fact that p c p. Lemma 1. Every minimum routing (at least one exists) is optimal. A greedy move is available to player i if she can obtain a lower path congestion by changing her current path from p i to p i the greedy move takes the original routing (p i ; p i ) to (p i ; p i ) in which p i is replaced by p i. Lemma 2. If a greedy move by any player takes p to p, then p < c p. 5

Thus, a greedy move decreases the number of high congestion edges, by transferring the congestion to lower congestion edges. Since there are only a finite number of routings, every best response dynamic is finite. By Lemma 2, no player can have an available greedy move at a minimum routing, as this would contradict the minimality of the routing. Hence, Lemma 3. Every minimum routing is an optimal Nash-routing. Hence, P os = 1. Theorem 1 now follows from Lemmas 2 and 3. 4 Path Length Bound on Price of Anarchy Here, we prove Theorem 2. In order to do so we will use the edge-expansion process, that we introduce here. Before we describe this technique we need to give some necessary definitions. Let R = (N, G, {P i } i N ) be an instance of a routing game. Let P = i N P i. The path-length of R is l = max p P p. A path-cut for player i is a set of edges E i such that every path in P i must use at least one of the edges in E i. The congestion of a path-cut C(E i ) is the minimum congestion of any edge in E i, C(E i ) = min e Ei C e. If player i is locally optimal with congestion pc i, then every alternative path for that player must have congestion at least pc i 1. Lemma 4. Let p = [p 1, p 2,, p N ] be a routing for which player i is locally optimal. Then, there is a path-cut E i for player i with congestion C(E i ) pc i 1. 4.1 Edge-Expansion Process If only some players are locally optimal in a routing p, then p is a partial Nashrouting (a Nash-routing is a special case of a partial Nash-routing). The edge expansion process applies to any partial Nash-routing. Suppose routing p has network congestion C, and suppose that at least one player is locally optimal with player cost C. Let E 0 be the set of edges with congestion C 0 = C that are used by at least one locally optimal player, and let Π 0 be the set of these locally optimal players that use at least one edge in E 0. By Lemma 4, each player in Π 0 has a path-cut with congestion at least C 0 1. Let E 1 denote the union of E 0 with all these path-cuts of every player in Π 0. Thus, E 0 E 1 and each edge in E 1 has congestion at least C 1 = C 0 1. Let Π 1 denote the set of locally optimal players whose paths in p use at least one edge in E 1. Note that Π 0 Π 1. Each player in Π 1 has player cost at least C 1, since every edge in E 1 has congestion at least C 1. We repeat this process as follows. Suppose that for i 1, edge set E i has been constructed as the union of E i 1 with path cuts for the players in Π i 1, thus every edge in E i has congestion at least C i = C i 1 1 = C i. We now construct Π i, the set of locally optimal players whose paths use at least one edge in E i ; every player in Π i has player cost at least C i. By Lemma 4, each player in Π i has a path-cut with congestion C i 1, and we construct E i+1 to be the union of E i with all these path-cuts of the players in Π i. 6

Using this inductive construction, we obtain a sequence of edge sets, E 0 E 1 E 2,, with C(E j ) C j = C j, and corresponding to each edge set, a set of locally optimal players Π 0 Π 1 Π 2. We continue this inductive construction up to edge set E s which is the first set for which E s 2 E s 1. We will refer to this process as the edge-expansion process. 4.2 Edge-Expansion Properties Since E i 1 2 n2 and each expansion at least doubles the size of the edge set, Lemma 5. E s 2 s 1 and 1 s < 2 log n. In routing p, let F (C ) N denote the set of non-locally optimal players with player cost at least C. We now establish a relationship between the congestion of a partial Nash-routing and the optimal routing. Lemma 6. C < 2l (C + F (C 2 log n)) + 2 log n. Proof. From the edge-expansion process, each edge in E s 1 has congestion at least C s 1. Let M be the number of times edges in E s 1 are used by the paths in p. Then, M > C s 1 E s 1. By construction, in p, the congestion in each of the edges of E s 1 is caused only by the players in A = Π s 1 B, where B F (C s 1 ) contains the non locally optimal players that use edges in E s 1. Since path lengths are at most l, each player in A can use at most l edges in E s 1. Hence, C s 1 E s 1 < M l A. Since, A Π s 1 + F (C s 1 ), we obtain, C s 1 < l E ( Π s 1 s 1 + F (C s 1 ) ). We now bound Π s 1. E s contains a path-cut for every player in Π s 1, and every such player must use at least one edge in E s in any routing, including the optimal routing p. Thus, edges in E s are used at least Π s 1 times, hence some edge is used at least Π s 1 / E s times, by the pigeonhole principle. Hence, C Π s 1 / E s (note that E s > 0). By the definition of s, E s 2 E s 1. Hence, Π s 1 2 E s 1 C, and C s 1 < 2l ( C + F (Cs 1) 2 E s 1 (Lemma 5), we obtain C < 2l ). Since C s 1 = C (s 1) and 2 E s 1 2 s ( ) C F (C s+1) + 2 + s 1. To conclude, 2 s 2, s and note that C < C implies F (C ) F (C ), hence F (C ) is non-increasing in C. Thus F (C s + 1) F (C 2 log n). Since in a Nash-routing, F (C ) = 0, C Lemma 6 with C, we obtain Theorem 2. > 0, by dividing the result of 5 Basic Cycle Bounds on Price of Anarchy Here, we first give the lower bound (part i) of Theorem 3 for the price of anarchy; we then prove the upper bound (part ii) of Theorem 3, for the special case of 2-connected graphs. The next result establishes the lower bound of Theorem 3. Lemma 7. For any graph G, there is a routing game with P oa κ e (G) 1. 7

Proof. Let Q = e 1,..., e κe be an edge simple cycle with length κ e. We construct a routing game with κ e players, where player i corresponds to edge e i = (u i, v i ) in Q, that is, the source of i is s i = u i and the destination t i = v i. The strategy set of i is the collection of all edge simple paths from s i to t i. There are two special paths in the strategy set of player i, the forward path which is composed solely of the edge (u i, v i ), and the backward path which consists of the remaining edges of cycle Q. Since Q is edge simple, if every player uses his forward path C = 1. Thus, the optimal social cost is 1. If on the other hand, all the players use their backward paths (backward routing p), then player i uses every edge in Q except e i exactly once. Thus, the congestion on every edge in Q is N 1 = κ e 1. Hence, if p is a Nash-routing, then P oa κ e 1. We will show that p is a Nash-routing by contradiction. Suppose that some player k is not locally optimal so player k has lower congestion for some other path p. Since every edge on Q has congestion κ e 1 in routing p, at least κ e 2 players other than player k use every edge on Q. Thus, if p uses any edge on Q, then pc k (p; p k ) = κ e 1, which does not improve its cost, so we conclude that p does not use any edge on Q. Therefore, p has length at least 2 (since p e k and G is not a multi-graph). Thus, replacing e k Q by p results in a new edge simple cycle Q that is strictly longer than Q, a contradiction. Thus, p is a Nash-routing. We now continue with the upper bound on the price of anarchy. A graph G is k-connected if its minimum edge-cut has size at least k. By Menger s theorem [32], G is k-connected if and only if there are at least k edge-disjoint paths between every two nodes. Let L be the longest path length in G. Lemma 8. If G is 2-connected, then κ e (G) 2L 3 2. The proof relies on the observation that the longest path p must have at least L edges in common with the largest cycle q, since otherwise, we would be able to construct a larger cycle by combing pieces of p and q. Lemma 8 bounds the longest path length in G with respect to κ e (G). Theorem 2 bounds the price of anarchy in terms of the longest path l in the players strategy sets. Since l L, we obtain the following result, with proves the upper bound of Theorem 3 for 2-connected graphs: Lemma 9. For any routing game on a 2-connected graph G, P oa c(κ e 2 (G)+ log n), for some constant c. 6 Cycle Upper Bound for General Graphs We now prove the upper bound (part ii) of Theorem 3 for general graphs. We will bound the price of anarchy with respect to the square of the longest cycle. The main idea behind the result is that any Nash-routing in G can be mapped to a partial Nash-routing on some 2-connected subgraph of G. In this partial Nash-routing, many players are locally optimal, and we can apply Lemma 6 in combination with Lemma 7 to obtain the result. 8

6.1 Canonical Subgraphs Consider an arbitrary connected graph G = (V, E). A subgraph G = (V, E ) of G contains a subset of the nodes, V V, and a subset of the edges E E, where each edge in E is incident with two nodes in V. We say that G is an induced subgraph by the node set V if E contains all the edges in E that are incident with a pair of vertices in V. We say that two subgraphs are adjacent if the intersection of their node sets is non-empty. The union of two subgraphs G = (V, E ) and G = (V, E ) is Ĝ = (V V, E E ). We will focus on 2-connected subgraphs. It is easy to verify that G contains a 2-connected subgraph if and only if it is not a tree. A 2-connected subgraph G is maximal if there is no larger 2-connected subgraph G = (V, E ) that contains G, so if G is 2-connected, then E E. Let A 1,..., A α be all the maximal 2-connected subgraphs of G, where α 1, and A i = (V Ai, E Ai ). Any two subgraphs A i and A j, i j, are node-disjoint since otherwise their union would be 2-connected, which contradicts their maximality. Therefore, we can construct from G two subgraphs A and B, where A consists of A 1,..., A α, while B consists of the remaining edges in G: A = (V A, E A ) and B = (V B, E B ), where E A = α i=1 E A i, E B = E E A, and V A and V B are the nodes adjacent to the edges in E A and E B, respectively. Note that graphs A and B are edge-disjoint, however, they may have common nodes. Subgraph B consists also of one or more disjoint maximal connected components (each containing at least two nodes), which we will denote B 1,..., B β. (Graph A consists of connected components A 1,..., A α.) We refer to the A i as the type-a canonical subgraphs of G and the B i as the type-b canonical subgraphs of G. One can show: Lemma 10. Every type-b subgraph is a tree. Any pair of type-a and type-b subgraphs can have at most one common node. We now define a simple bipartite graph H = (V H, E H ) that represents the structure of G. In V H = {a 1,..., a α, b 1,..., b β }, the nodes a i, b j correspond to the the type-a canonical subgraph A i and the type-b canonical subgraph B j respectively. The edge (a i, b j ) E H if and only if the canonical subgraphs A i and B j are adjacent (have a common node). The bipartition for H is (A, B), where A = {a 1,..., a α } and B = {b 1,..., b β }. The nodes in H inherit the same type as their corresponding canonical subgraph in G. Since G is connected, it follows immediately that H is connected too. Further, we have: Lemma 11. Graph H is a tree. 6.2 Canonical Subpaths A node in G can belong to at most one type-a subgraph and one type-b subgraph, since no two canonical subgraphs of the same type are adjacent. If a node is a member of one canonical subgraph, then its type is the type of the subgraph. If the node belongs to two canonical subgraphs then it is of type-a (we assign 9

it to the type-a canonical subgraph). An edge belongs to exactly one canonical subgraph and inherits the type of that subgraph. Let p = v 1, v 2,..., v k, k > 1, be an edge-simple path in G. We can write p as a concatenation of subpaths p = q 1 q 2 q k, where q i > 0, i, with the following properties: (i) the subpaths are edge disjoint; (ii) all the nodes of a subpath q i are in the same subgraph and have the same type, which will also be the type and subgraph of q i ; (iii) the types of the subpaths alternate, i.e. the types of q i and q i+1 are different; (iv) There is no type-a subpath with one node (any type-a subpath with one node can be merged with two adjacent type-b subpaths in the same type-b subgraph). We refer to the q i as the canonical subpaths of p. Note that there is a unique canonical subpath decomposition for path p. Since type-b subgraphs are trees and graph H is a tree, an arbitrary path in G can form cycles only inside type-a canonical subgraphs (in the respective type-a canonical subpaths). As a consequence, a path from a source node to a destination node follows a unique sequence of type-b edges (the union of all the edges in the type-b subpaths). Thus, we can obtain the following crucial result on paths that connect the same endpoints in G. Lemma 12. Any two edge-simple paths from nodes s to t in G use the same sequence of type-b edges. 6.3 Subgames in Canonical Subgraphs Consider a routing game R = (N, G, {P i } i N ) in G. Let p be a routing with network congestion C. Let p denote an optimal routing for R with congestion C. An immediate consequence of Lemma 12 is that every path in p uses the same type-b edges as its corresponding path in p, hence Lemma 13. Any type-b edge e has the same congestion in p and p, i.e. C e (p) = C e (p ) C. By Lemma 13, all the edges in p with congestion higher that C must occur in type-a subpaths. Lemma 14. For path p, if C p (p) > C, then p must have a type-a subpath q with C q (p) = C p (p). Suppose now that p is an arbitrary Nash-routing which has network congestion C. For a type-a subgraph Λ, let p Λ = {p 1,..., p γ } denote the paths in p that use edges in Λ, and denote the respective users as N Λ, where N Λ = γ. Let Q Λ = {q 1,..., q γ } denote the type-a canonical subpaths of the paths in p Λ that are in Λ (q i is a subpath of p i ). In subgraph Λ, we define a new routing game R Λ = (N Λ, Λ, {Pi Λ} i N Λ ), where Pi Λ contains all the type-a subpaths of P i that are in Λ and have the same source and destination as q i. We refer to R Λ as the subgame of R for subgraph Λ. Q Λ is a possible routing for R Λ. If q i is locally optimal for player i in Λ, we say that its corresponding path p i in G is satisfied in subgame R Λ. In other words, if path p i is satisfied in R Λ, player i does not wish to change 10

the choice q i in Λ. Every player with high player cost (higher than C ) must be satisfied in a type-a subgraph, since otherwise it would violate Lemma 14. Thus: Lemma 15. If player i has path p i and pc i > C, then player i is satisfied in some subgame R Λ in a type-a subgraph Λ, and player i has congestion pc i in Λ. 6.4 Main Result Consider routing game R = (N, G, {P i } i N ) in G and a Nash-routing p with congestion C(p) = C. Lemma 15, implies that each user is satisfied in some type-a subgraph (not necessarily the same). In any type-a subgraph, the resulting routing in the subgame may be a partial Nash-routing, since some users may not be satisfied in it. We first show that there is a subgraph with high congestion where the number of unsatisfied players is bounded. For a canonical type-a subgraph Λ, let F Λ (C ) denote the set of non-locally optimal players in the subgame R Λ whose congestion in R is at least C. We will use C Λ to denote the congestion in the canonical subgraph Λ. We have: Lemma 16. Suppose that C(p) > C + x(1 + log n) for some x > 0. Then, there is a type-a canonical subgraph Λ with congestion C Λ C x log n and F Λ (C Λ x) 2C. By combining Lemma 6 and Lemma 16 we obtain the following result which establishes the upper bound of Theorem 3. Lemma 17. P oa c (κ e 2 (G) + log 2 n), for some constant c. Proof. Let x = 2 log n. If C C + x(1 + log n), then there is nothing to prove because C/C 1 + 2 log n(1 + log n)/c c log 2 n, for some generic constant c. So, suppose that C > C + x(1 + log n). By Lemma 16, there exists a type-a subgraph Λ such that C Λ C 2 log 2 n and F Λ (C Λ 2 log n) 2C. By applying Theorem 6 to the subgame R Λ we obtain, C Λ < 2l (C Λ + F Λ (C Λ 2 log n )) + 2 log n, where l is the length of the longest edge-simple path in the player strategy sets in R Λ, n is the number of nodes in Λ and CΛ is the optimal congestion for the subgame R Λ. Note that n n, and the subgame R Λ cannot have a higher optimal congestion than the full game R, hence C CΛ. Since F Λ is monotonically non-increasing (F Λ (C ) F Λ (C ) for C < C ), we have that: C 2 log 2 n < 2l (C + F (C Λ 2 log n)) + 2 log n 2l (C + 2C ) + 2 log n. From Lemma 8, l cκ e 2 (Λ) cκ e 2 (G), and so C c (κ e 2 (G)C + log 2 n). After dividing by C, we obtain the desired result. 11

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